Q1 [2]. Vectors, hyperplanes, projections Note: in solving the math questions, aim for general (symbolic) solutions and substitute the specific numbers at the end.

This demonstrates a solid understanding of the key concepts.

Consider a vector space of two dimensions (x1, x2), a point A = (0, 1) and a vector v = (1, 1). a) What is the point defined by v considered as a position vector? If you move by one unit of length

from point A in the direction of v, what is the new point B you will arrive at?

b) What is the position vector of a point P derived by moving from point A along v by a distance s? The result is a parametric representation of a line, where the parameter is s. The line contains A and it is parallel to vector v.

c) Find vector u that is perpendicular to vector v above.

d) Given point A and vector u, provide a vector equation that every point P on the line must satisfy. Reduce this equation to the form ax1 + bx2 = 1, i.e. calculate a and b in terms of A and u. x1, x2 are the coordinates of P. Hint: the inner product of vector u and a vector parallel to the line is zero.

e) Generalize part (b) to the case of a plane in three dimensions. In this case, point A is a point in three dimensions. v will be replaced by two non-parallel vectors v, w that are both parallel to the plane. Complete this description.

f) Find vector u which is perpendicular to both v and w. Hint.

g) Generalize part (d) to a plane in three dimensions, i.e. given point A and vector u, provide a vector equation that every point P on the plane must satisfy. Can you reduce this equation to the form ax1+bx2+cx3 =1,wherex1,x2,x3 arethecoordinatesofP? Hint: the inner product of vector u with a vector parallel to the plane is zero.

h) Consider a plane in three dimensions defined by a point A and a normal vector u. Given a point B not on the plane, find the projection B of point B onto the plane. Hint: Come up with two properties of the projection B. One property is based on vector B A being parallel to the plane. Another property is based on vector B B being parallel to vector u.